Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C27 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C27 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
10474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,19,20,17)(16,18);(5,7)(6,9)(8,12)(10,11);(1,2,3,4)(7,12)(9,11)(13,15)(14,17)(16,20)(18,19);(6,10)(9,11)Polycyclic presentation:
2^9|a^2=g,c^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72217666, #72217699, #72217818, … |
| maximal normal subgroup of | #72217666, #72217699, #72217818, … |
| maximal quotient of | #72219805, #72219806, #72225787, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263109);